Question: Determine how many solutions exist for the system of equations. ${15x+3y = -15}$ ${y = -5-5x}$
Solution: Convert both equations to slope-intercept form: ${15x+3y = -15}$ $15x{-15x} + 3y = -15{-15x}$ $3y = -15-15x$ $y = -5-5x$ ${y = -5x-5}$ ${y = -5-5x}$ ${y = -5x-5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -5x-5}$ ${y = -5x-5}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${15x+3y = -15}$ is also a solution of ${y = -5-5x}$, there are infinitely many solutions.